Post on 25-Aug-2018
ECE 1311ECE 1311First Order Circuits Chapter 7
Dr. AHM Zahirul AlamECE Dept., IIUM
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The Step-Response of a RC Circuit (2)The Step Response of a RC Circuit (2)
Integrating both sides and considering the initial g g gconditions, the solution of the equation is:
⎩⎨⎧
>−+
<=
− 0)(
0)( /
0
0
teVVV
tVtv t
ssτ
Final value at Initial value at Source-freeFinal value at t -> ∞
Initial value at t = 0
Source free Response
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Complete Response = Natural response + Forced Response(stored energy) (independent source)
= V0e–t/τ + Vs(1–e–t/τ)
The Step-Response of a RC Circuit (3)p p ( )
Three steps to find out the step response of anThree steps to find out the step response of an RC circuit:
1. The initial capacitor voltage v(0).1. The initial capacitor voltage v(0).2. The final capacitor voltage v(∞) — DC voltage
across C.3. The time constant τ.
/τ/)]( )0([ )( )( tevvvtv −∞−++∞=Note: The above method is a short cut method You may also determine the
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Note: The above method is a short-cut method. You may also determine the solution by setting up the circuit formula directly using KCL, KVL , ohms law, capacitor and inductor VI laws.
The Step-Response of a RC Circuit (4)
Example 5
p p ( )
Find v(t) for t > 0 in the circuit in below. Assume the switch has been open for a long time and is closed at t = 0= 0. Calculate v(t) at t = 0.5.
2t
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Answer: and v(0.5) = V5060)( 2 −= − tetv
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The Step-response of a RL Circuit (1)p p ( )
The step response of a circuit is its behavior when the excitation is the step function, which may be a voltage or a current source.
• Initial currenti(0-) = i(0+) = Io
• Final inductor currenti(∞) = Vs/R
• Time constant τ = L/R
)()()( tueRVI
RVti
ts
os τ
−−+=
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)()()(RR o
The Step-Response of a RL Circuit (2)p p ( )
Three steps to find out the step response of anThree steps to find out the step response of an RL circuit:
1 Th i iti l i d t t i(0) t t 0+1. The initial inductor current i(0) at t = 0+.2. The final inductor current i(∞).3 The time constant3. The time constant τ.
τ/)]()0([)()( teiiiti −∞−++∞=
Note: The above method is a short cut method You may also determine
)]( )0([)( )( eiiiti ∞++∞
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Note: The above method is a short-cut method. You may also determine the solution by setting up the circuit formula directly using KCL, KVL , ohms law, capacitor and inductor VI laws.
The Step-Response of a RL Circuit (4)
Example 6
p p ( )
The switch in the circuit shown below has been closed for a long time. It opens at t = 0. Find i(t) for t > 0.
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•Read Chapters 6
•Required Problems -Ch. 6: 17, 46, 48, 54, 64
•Recommended Problems Ch 6: 13 19 53 65 73 -Ch. 6: 13, 19, 53, 65, 73
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Read Chapters 9 & 10 Appendix B Read Chapters 10 & 11Appendix B Required Problems
Ch. 7: 47, 64, 69 Ch. 9: 8, 11, 14, 16, 17 18
Read Chapters 10 & 11 Required Problems
Ch. 10: 19, 36, 42, 56, 80
R d d P bl17, 18
Recommended Problems Ch. 7: 50, 57, 71
Recommended Problems Ch. 10: 16, 39, 66, 77, 76
, ,Ch. 9: 9, 10, 19
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